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and in the long thin tongue of cold water extending far to the east, but we do nothave enough observational material to discuss this variability.The only other place in the world ocean where such low surface temperatures are

found at such low latitudes is the Peruvian coast, and even there it is uncommon tofind surface temperatures lower than 15'C within 100 of the equator. Such coldregions are often attributed to transport of surface waters away from the coast dueto local winds. The southwest monsoon is in proper direction to act in such amanner along this part of the Somali coast, but it seems to us that the intensity andlocation of the cold region is also to a large extent determined by the steep to-pographyassociated with thewestern boundary current itself-and this, of course, is aresult of wind distribution over the entire ocean, not merely locally. The relativeimportance of the two influences is not clear.

The authors are grateful to Drs. R. Currie, J. Swallow, and J. C. Crease aboard the Discov-ery for their collaboration on this survey, to Mrs. Mary Swallow for her constant interest, tothe National Science Foundation and the Office of Naval Research for financial support, and tothe officers, crew, and scientific parties of both vessels.

1 Greenspan, H., these PROCEEDINGS, 48, 2034 (1962); and J. Marine Res., 21, 147 (1963).2 Foxton, P., Deep-Sea Res., 12, 17 (1965).

STATISTICAL MECHANICS AND IRREVERSIBILITY*

BY FRANK C. ANDREWStTHEORETICAL CHEMISTRY INSTITUTE, UNIVERSITY OF WISCONSIN, MADISON

Communicated by Joseph Hirschfelder, May 13, 1965

Perhaps the most persistent truth the universe has thrust upon man is the factthat natural processes usually go spontaneously in only one direction. Most eventscannot be simply undone without changing yet other things; processes are usuallyirreversible. As the molecular theory of matter became accepted during the lasthundred years, there fell to the physicist the task of showing that all the physicalphenomena of nature followed as logical consequences of the fact that multitudes ofmolecules were moving and interacting according to mechanical laws. In partic-ular, the fundamental property of irreversible behavior was to be deduced from themechanics of the particles comprising matter.The foundation of a mechanical theory of irreversibility in classical gases was laid

by Maxwell in 1866.1 Based on this, in 1872 Boltzmann proved his H-theorem,2showing that the velocity distribution function approached equilibrium in a gasdescribed by the Boltzmann equation. Objections were immediately raised byLoschmidt,3 among others, who pointed out that the equations of motion of theparticles are completely reversible in time. This means that if a mechanical systemmay evolve from one state to another through a series of intermediate states, itcould also evolve from the final state to the initial, going through the intermediatesin the opposite order. If there is a certain probability of the evolution of a non-equilibrium system into one called "equilibrium," there must be a corresponding

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probability that the equilibrium system will spontaneously depart from equilibriumalong the reverse trajectory.

Further objections to Boltzmann's theory were based on Poincare's proof4 of thefact that a finite, conservative mechanical system initially in a given state willthrough its natural evolution return arbitrarily near to its original state, providedonly that a long enough time is allowed. This proof was considered by Zermelo,5among others, to show first that Boltzmann's theory must be wrong and, second, thatany kinetic theory must predict almost periodic behavior for a system, rather thanthe observed irreversible behavior. These are telling arguments, which have sur-vived to this day.Boltzmann fought back with every possible argument.6 He recognized the

naivete of any viewpoint holding that the distribution functions of kinetic theoryactually represented the distribution of velocities in the gaseous system. He con-sidered only that they represented the "most probable" distribution of velocitiesfound among the members of a statistical ensemble. Although Boltzmann's argu-ments have been repeated to this day, they have not proved satisfying, as the quota-tions below will show.The idea of an ensemble was stressed also by Gibbs,7 but in a different way from

Boltzmann. Gibbs viewed the equilibrium distribution function as describing theexact structure of the ensemble which represents an equilibrium system. Unfor-tunately, Gibbs' Chapter 12 on nonequilibrium did not clarify things much. Intheir famous Encyclopadie article,8 P. Ehrenfest and T. Ehrenfest concluded thatBoltzmann's views led to greater understanding than Gibbs'. However, Tolman,in his important book,9 sided with Gibbs. We note here simply that Gibbs' ideasare needed for treating nonideal gases and liquids, and we see no reason why theideal gas limit should involve a different concept than real gases.Innumerable discussions of the problem have appeared since, including several

comprehensive reviews.10-12 It is interesting to quote from books on statisticalmechanics to show how confusing the literature has been to readers who hoped thatthe molecular theory could shed light on the observed irreversible behavior in theuniverse: Jeans,13 after discussing the H-theorem in detail, concluded, "The motionis, in point of fact, strictly reversible, and the apparent irreversibility is merely anillusion introduced by the imperfections of the statistical method." Rushbrooke14states that it is "a false, indeed (bearing in mind the 'reversible' nature of theequations of mechanics and the 'directional' content of the second law of thermo-dynamics) an unreasonable, hope" to show the logical necessity of the approach toequilibrium through mechanics.

In the words of Kac,'5 "there are broadly speaking two problems: I. Is it pos-sible to reconcile both time reversibility and recurrence with 'observable' irreversiblebehavior? II. Is it possible to achieve such a reconciliation in the realm of classicalmechanics?" It is the hope of the author to present here a viewpoint and a set ofdefinitions that are consistent and logical and that show that both of Kac's questionsmay be answered "yes"! The presentation is based on the conclusions of a previouspaper. 16

First, it is important to agree on what problem statistical mechanics sets aboutsolving. In the author's opinion, that problem is the following:'7 There is a phys-ical system about which, as a result of measurements, certain information is known.

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Using this information and the fact that the system evolves through the mechanicalmotion of its constituent particles, it is the task of statistical mechanics to computethe probabilities of the various possible results of subsequent measurements onemight choose to make on the system. It would be unscientific to claim any morethan that of statistical mechanics. There is insufficient information available todetermine, even in principle, the mechanical evolution of the system. Thus, onlyprobabilities are accessible. Of course, if a particular result proves overwhelminglyprobable, it may be predicted with an appropriate degree of confidence. Use ofprobability theory requires the establishment of an ensemble to represent thesystem (ref. 9, chaps. III and IX; ref. 17, § 4). The probability of a particularresult of a measurement on the system is by definition, then, the fraction of membersof the ensemble which would yield that result for that measurement.Any problem in statistical mechanics, equilibrium or nonequilibrium, can be

separated into three parts: First, one finds the correct way to construct the initialensemble, building into it the information available. Second, if the problem is non-equilibrium, one follows the mechanical evolution of the ensemble for the requiredperiod of time. This is possible in principle with classical mechanics, since eachmember of the ensemble is initially in a completely specified state. With quantummechanics, only transition probabilities may be used. This part of the problemmay be omitted for equilibrium problems, since for them one assures one's self thatall probabilities generated by the ensemble are time-independent. Third, onecomputes the expectation values (e.g., ensemble averages) of whatever physicalquantities are of interest, and one may, if one wishes, determine the likelihood ofsignificant fluctuations or deviations from these predicted values. This procedurederives maximum utility from the available information.The first step is to construct the initial ensemble to reflect the available informa-

tion and none other (ref. 9, chap. IX). For a quantum-mechanical problem onemerely has to ask which of the allowed quantum states for the system are consistentwith the initial information. Since there is no further information permitting apreference for any of these states over any of the others, each of them must beweighted equally. It is intuitively obvious that any other way of constructing theensemble would yield a distribution of probabilities which was a function of some-thing other than the available information. It would be arbitrary and capriciousto make the ensemble reflect such additional, spurious information. In assigningphases to the wave functions for members of the ensemble, similar argumentsdemand that if one's information is such that no range of phases should be preferredover any other, then they should be randomly weighted between 0 and 2w (an exten-sive discussion is given in ref. 9, chap. IX). A study of this problem by the disciplineof information theory18 concurs with these conclusions. If a classical ensemble is tobe constructed, it should represent the classical limit of the appropriate quantumensemble. Because of the uncountable infinity of possible classical states, classicalensembles have an uncountable infinity of members (except for some special, un-physical model-problems).The ensemble one constructs thus reflects one's knowledge about the system.

For example, the condition of equilibrium is one of maximum ignorance. If asystem under given constraints is "in equilibrium at a given temperature" or "inequilibrium with a given energy," then all one knows is that the system with those

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constraints has sat around long enough that any other information one might havehad about it has become meaningless. This is a time-independent condition, andthe resulting distribution function must be time-independent. It must correctlygive not only the values of the most probable results of measurements, but also theprobabilities of unlikely results, even ones which the measurer would call "wildlyunlikely fluctuations." Any distribution function in statistical mechanics containsthe probabilities of all possible fluctuations consistent with the information. Fluc-tuations are part of any macroscopic condition, equilibrium included.The second part of the problem in nonequilibrium statistical mechanics is to

remove some constraint or "turn on the time" and apply the known mechanicallaws to each member of the ensemble. The structure of the ensemble and con-sequently the properties predicted for the system change with time in a way thatreflects the logical consequences of the initial information about the condition of thesystem and our knowledge of the mechanical laws by which the system evolves.This part of the problem occupies most writers on nonequilibrium statistical me-chanics; usually a number of approximations must be made to simplify things; andthese approximations are often blamed for introducing irreversibility. The authorhas indicated in a previous paper,'6 however, that even if this part of the problem istreated exactly, the distribution functions will change with time in a completelyirreversible fashion and the predicted values of all interesting properties of thesystem will reach their equilibrium values and stay there forever. The only excep-tions are a few artificial, unphysical models which possess both the following features:(1) The microscopic mechanics which underlies the macroscopic properties is deter-ministic. (2) The information initially available about the system is such thatonly a finite number of choices of the phase space variables of the microscopicmechanics is consistent with it. Exactly calculable examples are given by theauthor which illustrate this conclusion. Thus, even finite classical mechanicalsystems yield truly irreversible statistical mechanics whenever an infinite numberof classical states is consistent with the initial information.

It is true that the approximations made in this second part of the problem usuallyresult in loss of information. But it is a very poor choice of definitions to equateinformation loss with irreversibility, since the quality of a piece of information maychange drastically while its quantity remains fixed. After information becomesphysically unimportant, it may be thrown out.Use of an example is perhaps the best way to clarify the situation: Consider two

bottles connected by a stopcock, bottle 2 initially evacuated and bottle 1 containinga gas. At time zero the stopcock is opened and the gas rushes from 1 into 2. Ac-cording to classical thermodynamics, the properties of a system are characterizedby its energy and constraints. Therefore, once the stopcock is opened, the onlyvalue thermodynamics predicts for, say, the density is the final "equilibrium" value.Irreversibility or the increase of entropy arises from the loss of the informationabout which bottle contains the gas. The moment the stopcock is opened, classicalthermodynamics can no longer predict where the gas will be, so the entropy issuddenly increased.

For density to be predicted as a function of time in some part of the system, theactual mechanics of the particles is involved. For this, statistical mechanics isneeded. If classical mechanics is being used, the initial distribution function

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describes an equilibrium with all N particles in bottle 1, either a microcanonical orcanonical ensemble depending on whether E or T is known. When the stopcock isopened, such a distribution is no longer time-independent. The evolution of fN,the complete distribution function, is given by the Liouville equation, with- appro-priate boundary conditions at the walls (ref. 9, chap. III). For some boundary con-ditions, such as those implying elastic collisions, fN is a constant of the motion, thusfN(t) reflects the reversibility of the mechanical equations by preserving forever theinformation that at t = 0 all particles were in bottle 1.

It is important to realize in just what sense this information is still present.Before the stopcock is opened, the information refers to individual particles: "Theyare all in bottle 1." One minute after the stopcock is opened, however, the in-formation has changed its character: "The N particles' positions and momenta arecorrelated in a fantastically complicated manner. Wherever the particles may bein the over-all system, their state must be such that if all momenta were simul-taneously reversed, in one minute allN particles would be in bottle 1." There is anenormous difference in physical accessibility of the two statements of informationalcontent. The first is immediately verifiable and of profound importance in predict-ing the physical properties of the system prior to the stopcock's being opened.The second is in no way verifiable, because the momenta cannot be reversed simul-taneously.'9 It has no importance whatever in predicting the physical propertiesone minute after opening the stopcock.Our example illustrates the nature of a true irreversibility in mechanics. Suppose

one could follow the evolution of the ensemble exactly. A reasonable value topredict for the number of particles in bottle 2 is the average over the ensemble.At t = 0 this average is zero. With time, the exact mechanical evolution of theensemble leads to predicting a steadily increasing number in bottle 2. Finally, thepredicted number reaches about N/2 and stays there. 16 This is a true irreversibilityin the results predicted for the observable properties of the system.We are now able to comment on a point regarding irreversibility made by Kac

(ref. 15, p. 85): "Gibbs's idea that probability should enter Mechanics only throughfN(O) is, of course, very appealing. In general, this view is probably untenable andprobability must be made to intervene in some other ways." We have, with Gibbs,introduced probability into mechanics through the construction of fN(O). In con-tradiction to Kac, we have noted, e.g., in the exactly calculable example of refer-ence,'6 that this is all the probability needed-fN(t) has irreversible behavior, despiteits preservation of the initial information and its being a constant of the motion.This is also in spite of the fact that the individual members of the ensemble areundergoing Poincar6 recurrences. We thus reach Gibbs' conclusion.However, we must observe that this conclusion is rather pedantic, because fN(t)

is not an observable quantity. Only ensemble averages of dynamical quantitiescorresponding to observable quantities are accessible. It is the true irreversibilityin these averages which bears stressing. And as soon as these averages are per-formed, probability has entered again (another coarse-graining, as some might say).It is, however, a completely natural and unavoidable entrance, which may perhapsbe best described by the following comments: The dynamical quantities whoseaverages yield predicted values of observable quantities involve only a few phasespace variables.20 Therefore, it is not the complete fN which is needed to compute,

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the averages, but a few low-order reduced distribution functions like fi and f2.These reduced distribution functions do not reflect complicated many-particlecorrelations. This shows the utter lack of physical significance of such correlations.During the evolution of the ensemble, as the initial information gets lodged in moreand more complex correlations involving more and more particles, it becomesincreasingly lost to fi and f2. Eventually, so far as the physically important aver-ages are concerned, the results calculated from the exact fN (or fi or f2) would differnegligibly from those found using an equilibrium distribution function.As a result of these conclusions, one can now go back and reconsider the first step

in the basic statistical mechanical problem construction of the initial ensemble.It is clearly unimportant that the entire.fN(O) reflect only the initial information.Spurious information can be permitted so long as its presence does not change thepredictions. In practice, this gives great latitude in the setting up of a problem.If the quantum phases are not correctly randomized or if some available quantumstates are omitted from the initial ensemble or if some unphysical many-particlecorrelation is built into .fN(O), the results are frequently unchanged. The construc-tion of fN(O) is thus an irrelevant question; only the low-order reduced distributionfunctions may need to be specified. However, they, of course, must be made toreflect only the correct information.

Since the ensemble does show irreversible behavior, it might be asked how fluctua-tions and Poincare cycles are possible. Certainly, the arguments based on thereversible nature of mechanics have not been eliminated. The answer is that fN(t)does not determine results of observations of the system. It does enable them to bepredicted, often with great confidence, and furthermore permits finding the prob-abilities of fluctuations of various magnitudes, including even such wild fluctuationsas are represented by IPoincar6 cycles. However, these probabilities will differnegligibly from those found from the equilibrium fN. Nevertheless, they will bethe best it is possible to calculate from the information at hand.The probabilities of significant fluctuations are vanishingly small whenever the

region involved contains an enormous number of particles. However, any con-ceivable fluctuation in any property, consistent with the initial information andthe mechanical motion, is represented in the ensemble and thus has nonzero prob-ability. So if one makes enough observations over a long enough time, one willobserve such fluctuations. The gas molecules in our example will all come backinto bottle 1. In this sense mechanics is strictly reversible. But there is by nomeans enough information to predict when this will happen. The equilibriumprobability of (V)N for all N molecules of a dilute gas to be in bottle 1 is about whatone is left with, despite the knowledge that they were all there previously at timezero. The system, being in some mechanical state, is actually represented by onlyone member of the ensemble. Since one does not know by which member, onemust base one's predictions on the entire ensemble. Statistical mechanics convincesthe prudent predictor to stay with the equilibrium values of the variables once theyhave been reached.

Thus, one may view irreversible equations in statistical mechanics, such asBoltzmann's equation for fi in a dilute gas, as representing to some degree of ap-proximation the way in which the reduced distribution function describing theensemble actually evolves. Equations for reduced distribution functions are more

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useful than ones for fN, because they focus attention on the meaningful correlationamong few particles that affects observable quantities. It is natural that theseequations be irreversible. Even fN behaves irreversibly, and the equations forreduced distribution functions must reflect not only that, but also the inevitable lossof information from fi and f2 to correlations involving more and more particles.The fi of Boltzmann's equation does not represent the distribution of individualparticles in the system, as a natve kinetic theory might suggest. Neither does itrepresent the most probable distribution of particles in members of the ensemble asBoltzmann suggested. Instead, f,(r,p) is simply N times2' the probability densitythat some particle, say, number 1, be at r and p. This is the same for any of thenumbered particles because fN(1,2, . . . ,N) is constructed to be unchanged uponpermutation of particles. It is this quantity, fA, describing the entire ensemble,which under specified conditions approximately satisfies Boltzmann's equation.A final observation may be made regarding the means by which a distinction

between past and future enters statistical mechanics." This is invariably intro-duced through a causality condition in the completely natural way: if two eventsare correlated, it is due to their interaction in the past. This is the way man alwaysdetermines the direction of time.

Before concluding, it is important to consider exceptions to the above conclusions.If a model being treated with deterministic mechanics has initial information whichallows only a finite number of mechanical states, fN(t) will be an almost periodicfunction of time.'6 Such a case would almost certainly imply that attention wasbeing focused from the beginning on the mechanical state of the system rather thanon its macrostate. Not enough probability could enter such a problem to give riseto irreversibility. There would be enough information to predict fluctuations.The above discussion does not explicitly mention the case in which a nonequilib-

rium system is allowed to evolve from time zero to T. at which time all the momentaare reversed. '9 In this case, one must construct the ensemble at time zero to reflectthe available information. Then at time r, all momenta in the ensemble must bereversed. The ensemble again gives the best possible predictions of what willhappen. It may well predict that during the period r to 2r the system will returnto its original unlikely condition. That should not cause surprise.

This paper has ignored the question of the ultimate loss of information at thesystem boundaries. This is not serious because once information becomes physi-cally inaccessible, it makes no difference whether or how it is lost from the completedistribution function. Certainly, the initial information is ultimately lost from areal system through interaction with the environment. Rarely, however, is thisinteraction important in governing the physical properties in the interior of a system.In those cases where system-wall interactions are so strong that they do affect thesystem properties, even classical thermodynamics breaks down. The nature of theboundary conditions then becomes critical. Otherwise it is important only to usea reasonable boundary condition that does not do violence to the physics of theproblem.The aim of this paper is to express what statistical mechanics can actually do and

to exclude claims for anything beyond its capabilities. The initial ensemble or dis-tribution function is constructed to contain only what information is known aboutthe system. The distribution function is then studied as a function of time. The

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strictly mechanical evolution within the members of the ensemble changes thecomplete distribution function and also the probabilities of various experimentalresults. Reduced distribution functions, from which experimental quantities maybe calculated, show the truly irreversible way in which excess initial informationbecomes inaccessible due to the mechanical motion of the particles. The predic-tions of statistical mechanics are driven irreversibly by the mechanics towardequilibrium values. On the other hand, the actual physical system most likelyevolves in a way described by ensemble averages over the reduced distribution func-tions. It may show fluctuations in its properties at any time, the probabilities ofvarious fluctuations also being calculable from the distribution functions. Largefluctuations are extremely improbable, due to the large number of particles in amacroscopic region. Thus, statistical mechanics cannot show the logical necessityof the approach to equilibrium in a mechanical system. That is gratifying, becausethe approach to equilibrium is not necessary, as Poincare proved. Statisticalmechanics can, however, show logically the overwhelming probability of theapproach to the almost constant values of physical properties that are usuallyassociated with the equilibrium condition.The author gratefully acknowledges helpful discussions with C. F. Curtiss on the subject of this

paper and also the interest of J. 0. Hirschfelder in this work.* Research supported in part by National Science Foundation grant NSF-G20725.t Alfred P. Sloan Fellow.1 Maxwell, J. C., Phil Mag., [4] 32, 390 (1866).2 Boltzmann, L., Wien. Ber., 66, 275 (1872).3 Loschmidt, J., Wien. Ber., 73, 128, 366 (1876); 75, 287 (1877); 76, 209 (1877).4 Poincare, H., Acta Math., 13, 1 (1890).5 Zermelo, E., Ann. Physik, [3] 57, 485 (1896); 59, 793 (1896).6 Boltzmann, L., Vorlesungen uiber Gastheorie (Leipzig; J. A. Barth, Part I, 1896; Part II,

1898). [English translation by S. G. Brush, Lectures on Gas Theory (Berkeley: Univ. of Calif.Press, 1964)].

7 Gibbs, J. W., Elementary Principles in Statistical Mechanics (New Haven, Connecticut:Yale Univ. Press, 1902).

8 Ehrenfest, P., and T. Ehrenfest, The Conceptual Foundations of the Statistical Approach inMechanics, 1912 [English translation by M. Moravcsik (Ithaca: Cornell Univ. Press, 1959)].

9 Tolman, R. C., The Principles of S.atistical Mechanics (London: Oxford Univ. Press, 1938).10 ter Haar, D., Revs. Modem Phys., 27, 289 (1955).Reichenbach, H., The Direction of Time (Berkeley: Univ. of California Press, 1956).

12 Uhlenbeck, G. E., in ref. 15, p. 183, and Lectures in Statistical Mechanics (Providence:American Mathematical Society, 1963).

13 Jeans, J. H., The Dynamical Theory of Gases (Cambridge Univ. Press, 4th ed., 1925, reprinted1954 by Dover Publications), chap. IV and p. 38.

14 Rushbrooke, G. S., Introduction to Statistical Mechanics (London: Oxford Univ. Press, 1949),p. 315.

16 Kac, M., Probability and Related Topics in Physical Sciences (New York: Interscience Pub-lishers, 1959), p. 73.

16 Andrews, F. C., these PROCEEDINGS, 53, 1284 (1965).17 Andrews, F. C., Equilibrium Statistical Mechanics (New York: John Wiley and Sons, 1963),

sect. 3.18 Brillouin, L., Science and Information Theory (New York: Academic Press, 1962), 2d ed.19 A not too analogous system for which all the momenta can be effectively reversed simul-

taneously is the spin-echo experiment [Hahn, E. L., Phys. Rev., 80, 580 (1950)].20 Irving, J. H., and J. G. Kirkwood, J. Chem. Phys., 18, 817 (1950).21 Different authors may use different normalizations.

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